Maths WorkshopMaths problems with full solutions

Matrices and linear systems

45 problems · Matrix operations, Gaussian elimination, matrix inverses and linear systems.

Ex.14Matrix addition and scalar multiplication
#matrix operations #addition #scalar multiplication
Matrices and linear systems
High school
High school
Ex.189Inverse of a 2×2 matrix
#inverse #determinants
Matrices and linear systems
High school
High school
Ex.327Inverting I − A
#matrices #nilpotent
Matrices and linear systems
L2
L2
Ex.328Kernel of a matrix
#kernel #rank
Matrices and linear systems
L1
L1
Ex.506Determinant of a 3×3 matrix by Sarrus' rule
#determinant #Sarrus rule #3x3 matrices
Matrices and linear systems
High school
High school
Ex.507Inverse of a 2×2 matrix
#inverse #determinant #2x2 matrices
Matrices and linear systems
L1
L1
Ex.508Matrix multiplication
#matrix multiplication #non-commutativity
Matrices and linear systems
High school
High school
Ex.509Powers of a diagonal matrix
#diagonal matrix #powers #induction
Matrices and linear systems
L1
L1
Ex.510Properties of the trace
#trace #cyclic property #similarity invariance
Matrices and linear systems
L1
L1
Ex.511Properties of the transpose
#transpose #properties #matrix operations
Matrices and linear systems
L1
L1
Ex.1048Block multiplication
#block matrices #multiplication #structure
Matrices and linear systems
L1
L1
Ex.1049Change of basis
#change of basis #transition matrix #similarity
Matrices and linear systems
L1
L1
Ex.1050Cramer's rule
#Cramer's rule #determinant #linear system
Matrices and linear systems
L1
L1
Ex.1051Determinant of a block triangular matrix
#block matrices #determinant #triangular structure
Matrices and linear systems
L1
L1
Ex.1052Using determinant properties
#determinant #row operations #properties
Matrices and linear systems
L1
L1
Ex.1053Eigenvalues and eigenvectors of a 2×2 matrix
#eigenvalues #eigenvectors #characteristic polynomial
Matrices and linear systems
L1
L1
Ex.1054Solving a linear system by Gauss-Jordan
#linear system #Gauss-Jordan #row reduction
Matrices and linear systems
High school
High school
Ex.1055Gram-Schmidt orthogonalization
#Gram-Schmidt #orthogonalization #orthonormal basis
Matrices and linear systems
L1
L1
Ex.1056Invertibility depending on a parameter
#invertibility #parameter #determinant
Matrices and linear systems
L1
L1
Ex.1057Kernel and image of a matrix
#kernel #image #basis
Matrices and linear systems
L1
L1
Ex.1058Rank-nullity theorem
#rank-nullity #kernel #image
Matrices and linear systems
L1
L1
Ex.1059Powers of a nilpotent matrix
#nilpotent #powers #induction
Matrices and linear systems
L1
L1
Ex.1060Computing the rank of a matrix
#rank #Gaussian elimination #row reduction
Matrices and linear systems
L1
L1
Ex.1061Similarity invariants
#similarity #characteristic polynomial #trace #determinant
Matrices and linear systems
L1
L1
Ex.1762The Cayley-Hamilton theorem
#Cayley-Hamilton #characteristic polynomial #matrix powers
Matrices and linear systems
L1
L1
Ex.1763Cholesky factorization of a 2×2 matrix
#Cholesky #positive definite #factorization
Matrices and linear systems
L1
L1
Ex.1764Diagonalizing a 2×2 matrix
#diagonalization #eigenvalues #change of basis
Matrices and linear systems
L1
L1
Ex.1765Diagonalizing a 3×3 matrix
#diagonalization #eigenvalues #eigenvectors
Matrices and linear systems
L1
L1
Ex.1766When is a matrix diagonalizable?
#diagonalization #eigenspaces #multiplicity
Matrices and linear systems
L1
L1
Ex.1767Eigenvalues of a 3×3 matrix
#eigenvalues #characteristic polynomial #trace and determinant
Matrices and linear systems
L1
L1
Ex.1768Inverse of a 3×3 matrix by row reduction
#inverse #Gaussian elimination #3x3 matrices
Matrices and linear systems
L1
L1
Ex.1769A linear recurrence and matrix powers
#linear recurrence #diagonalization #matrix powers
Matrices and linear systems
L1
L1
Ex.1770LU decomposition
#LU decomposition #Gaussian elimination #factorization
Matrices and linear systems
L1
L1
Ex.1771The minimal polynomial of a matrix
#minimal polynomial #characteristic polynomial #diagonalizability
Matrices and linear systems
L1
L1
Ex.1772Orthogonal projection onto a subspace
#orthogonal projection #projection matrix #orthonormal basis
Matrices and linear systems
L1
L1
Ex.1773Positive definiteness of a symmetric matrix
#positive definite #Sylvester criterion #eigenvalues
Matrices and linear systems
L1
L1
Ex.1774Powers of a companion matrix
#companion matrix #powers #recurrence
Matrices and linear systems
L1
L1
Ex.1775Reducing a quadratic form
#quadratic form #reduction #completing the square
Matrices and linear systems
L1
L1
Ex.1776Signature of a quadratic form
#quadratic form #signature #diagonalization
Matrices and linear systems
L1
L1
Ex.1777Rank depending on a parameter
#rank #parameter #Gaussian elimination
Matrices and linear systems
L1
L1
Ex.1778A linear system with a parameter
#linear system #parameter #discussion
Matrices and linear systems
L1
L1
Ex.2178Diagonalizability depending on a parameter
#diagonalization #parameter #characteristic polynomial
Matrices and linear systems
L1
L1
Ex.2179QR factorization by Gram-Schmidt
#QR factorization #Gram-Schmidt #orthogonalization
Matrices and linear systems
L1
L1
Ex.2180Singular value decomposition of a 2×2 matrix
#SVD #singular values #orthogonal matrices
Matrices and linear systems
L1
L1
Ex.2181Orthogonal diagonalization of a symmetric matrix
#symmetric matrix #orthogonal diagonalization #spectral theorem
Matrices and linear systems
L1
L1